A005785 Number of 5-covers of an unlabeled n-set. (Formerly M3949)

1, 5, 28, 156, 863, 4571, 22952, 108182, 477136, 1969270, 7625579, 27804973, 95858868, 313747418, 978734539, 2920530663, 8363945469, 23057872913, 61357278239, 157985305473, 394486861086, 957156158394, 2260761331227

Offset

0,2

Comments

Number of 5 X n binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Stefano Spezia, Table of n, a(n) for n = 0..10000

R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.

Vladeta Jovovic, Binary matrices up to row and column permutations

Index entries for linear recurrences with constant coefficients, signature (7, -17, 13, 7, -11, 16, -44, 43, -65, 82, 46, -36, -28, -175, 85, -168, 504, 91, 77, -394, -664, -52, -382, 1642, 600, 813, -1209, -1632, -1650, -1050, 2982, 2124, 3592, -1360, -2074, -5329, -3607, 1970, 3608, 7640, 1778, 426, -8168, -6638, -3524, 2095, 8401, 6077, 5907, -5907, -6077, -8401, -2095, 3524, 6638, 8168, -426, -1778, -7640, -3608, -1970, 3607, 5329, 2074, 1360, -3592, -2124, -2982, 1050, 1650, 1632, 1209, -813, -600, -1642, 382, 52, 664, 394, -77, -91, -504, 168, -85, 175, 28, 36, -46, -82, 65, -43, 44, -16, 11, -7, -13, 17, -7, 1).

Formula

G.f.: - (x^68 - 2*x^67 + 10*x^66 + 32*x^65 + 175*x^64 + 794*x^63 + 3441*x^62 + 13186*x^61 + 46027*x^60 + 146118*x^59 + 427347*x^58 + 1155432*x^57 + 2912873*x^56 + 6875608*x^55 + 15281029*x^54 + 32094658*x^53 + 63945531*x^52 + 121210914*x^51 + 219194198*x^50 + 378998758*x^49 + 627863648*x^48 + 998282344*x^47 + 1525746624*x^46 + 2244502676*x^45 + 3181886869*x^44 + 4351201210*x^43 + 5744918381*x^42 + 7328807372*x^41 + 9039504349*x^40 + 10785767638*x^39 + 12455264802*x^38 + 13925287384*x^37 + 15077477135*x^36 + 15812782150*x^35 + 16065602576*x^34 + 15812782150*x^33 + 15077477135*x^32 + 13925287384*x^31 + 12455264802*x^30 + 10785767638*x^29 + 9039504349*x^28 + 7328807372*x^27 + 5744918381*x^26 + 4351201210*x^25 + 3181886869*x^24 + 2244502676*x^23 + 1525746624*x^22 + 998282344*x^21 + 627863648*x^20 + 378998758*x^19 + 219194198*x^18 + 121210914*x^17 + 63945531*x^16 + 32094658*x^15 + 15281029*x^14 + 6875608*x^13 + 2912873*x^12 + 1155432*x^11 + 427347*x^10 + 146118*x^9 + 46027*x^8 + 13186*x^7 + 3441*x^6 + 794*x^5 + 175*x^4 + 32*x^3 + 10*x^2 - 2*x + 1)/((x^6 - 1)^2*(x^4 + x^3 + x^2 + x + 1)^6*(x^3 - x^2 + x - 1)^6*(x^2 + x + 1)^6*(x + 1)^10*(x - 1)^23).

a(n) = n^30/(30!*5!) + O(n^29). - Vaclav Kotesovec, Aug 09 2022

Prog

(PARI) Vec(G(5, x)*(1 - x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

Crossrefs

Column 5 of A055080.

First differences of A052264.

Cf. A002620, A005783, A005784, A055066.

Sequence in context: A164537 A015536 A271808 * A027912 A378462 A243669

Adjacent sequences: A005782 A005783 A005784 * A005786 A005787 A005788

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Jun 03 2000

a(0) = 1 prepended by Stefano Spezia, Aug 09 2022

Status

approved